Abstract

Future high resolution, high sensitivity Sunyaev-Zeldovich (SZ) observations of individual clusters will provide an exciting opportunity to answer specific questions about the dynamical state of the intra-cluster medium (ICM).
In this paper we develop a new method that clearly shows the connection of the SZ signal with the underlying cluster model. We include relativistic temperature and kinematic corrections in the single-scattering approximation, allowing studies of hot clusters. In our approach, particular moments of the temperature and velocity field along the line-of-sight determine the precise spectral shape and morphology of the SZ signal.
We illustrate how to apply our method to different cluster models, highlighting parameter degeneracies and instrumental effects that are important for interpreting future
high-resolution SZ data.
Our analysis shows that line-of-sight temperature variations can introduce significant biases in the derived SZ temperature and peculiar velocity. We furthermore discuss how the position of the SZ null is affected by the cluster’s temperature and velocity structure. Our computations indicate that the SZ signal around the null alone is rather insensitive to different cluster models and that high frequency channels add a large leverage in this respect.
We also apply our method to recent high sensitivity SZ data of the Bullet cluster, showing how the results can be linked to line-of-sight variations in the electron temperature.
The tools developed here as part of SZpack should be useful for analyzing high-resolution SZ data and computing SZ maps from simulated clusters.

These encouraging prospects also raise a number of important problems that must be addressed before the rich information contained in the future SZ data can be fully exploited.
One is simply related to the precise and fast computation of the SZ signal given basic parameters of the scattering medium, such as the Thomson scattering optical depth, τ, the electron temperature, Te, and bulk velocity, βc, while accounting for relativistic temperature and kinematic corrections.
Previously, this issue has been addressed by several groups (Rephaeli, 1995b; Challinor & Lasenby, 1998; Itoh et al., 1998; Sazonov & Sunyaev, 1998; Nozawa et al., 1998a; Challinor & Lasenby, 1999; Chluba et al., 2005) by means of Taylor expansions for the SZ signal. While the evaluation of these expansions is very fast, they are limited to rather low temperature gas (Fig. 2 shows that for Te≳13keV this approach breaks down).
One alternative is direct numerical integration of the Boltzmann collision term making use of the symmetries of the scattering problem (Wright, 1979; Dolgov et al., 2001; Nozawa et al., 2009; Poutanen & Vurm, 2010), but this is time-consuming and not well-suited for extensive parameter estimations or computations of the SZ signal from simulated clusters.
On the other hand, a fast but not as precise and flexible approach is simple tabulation of the SZ signal or the use of convenient fitting function (Nozawa et al., 2000; Itoh & Nozawa, 2004; Shimon & Rephaeli, 2004).

Recently, Chluba et al. (2012, CNSN in the following) developed a method in the middle of these extremes. In their work, a new set of frequency-dependent basis functions was computed numerically to allow very fast and precise calculation of the SZ signal. The basis functions are informed by the underlying physics of the scattering problem and thus are ideally suited for future SZ signal processing and parameter estimation.
The associated routines are part of SZpack5.
However, several extensions are required. Firstly, so far high precision (relative accuracy ≃0.001%) calculations with SZpack were limited to Te≲25keV.
This problem can be easily overcome using the method of CNSN by appropriate extensions of the basis functions, as we explain in Sect. 2.4.
With SZpack v1.1, which is presented here, it is now possible to compute the SZ signal for Te≲75keV and βc≲0.01 to ≃0.001% relative precision at practically no computational cost.
This precision and range of parameters covers all physically relevant cases and hence provides an important preparation for SZ parameter estimation without significant limitations.

Secondly, line-of-sight variations of the temperature and velocity field (with any of the aforementioned methods) can only be accounted for by means of additional 1-dimensional integrals; this again makes extensive SZ parameter estimation expensive.
Especially when computing the SZ signal from cluster simulations the problem becomes very demanding, even if evaluation for single gas parameters (τ, Te, and βc) is extremely fast.
Here we reformulate the representation of the SZ signal to overcome this limitation.
We utilize that in the single-scattering approximation, frequency-dependent terms can be separated from temperature- and velocity-dependent contributions (Sect. 2).
This implies that the SZ signal for a given cluster model can be calculated using appropriate moments of the temperature and velocity field.
While this means that a finite number of 1-dimensional integrals along different lines-of-sight have to be evaluated, this separation still greatly reduces the computational burden because afterward the SZ signal at any frequency can be computed as simple matrix multiplication.

While the method developed in Sect. 2 is both precise and fast, delivering quasi-exact results for the SZ signal through different lines-of-sight for any cluster atmosphere, in the future analysis of high resolution, high sensitivity SZ data another simplification is possible. In Sect. 3, we show that for typical cluster models the smoothness of the temperature and velocity profiles allows minimizing the number of parameters needed to accurately describe the SZ signal, resulting in a second set of moments that are related to the line-of-sight temperature and velocity dispersions and higher order statistics (see Eq. (18) for instance). The associated expansion of the SZ signal around the mean becomes perturbative and the number of moments needed to describe the SZ signal depends directly on the observational sensitivity.

Our formulation furthermore allows direct separation of frequency-dependent from spatially varying terms, providing a clear link between morphological changes of the SZ signal and cluster parameters.
For example, the presence of large-scale, post-merger cluster rotation can introduce a bipolar kSZ signal, which is related to a spatially varying average line-of-sight velocity. The superposition of thSZ with this rotational kSZ implies small frequency-dependent changes of the clusters morphology (Chluba & Mannheim, 2002). Similarly, variations of the electron temperature along the line-of-sight introduce morphological effects (as also pointed out more recently by Prokhorov et al., 2011), and as we explain here, spatial variations of temperature and velocity moments are the source of these morphological changes.
The moments therefore constitute the main observables of high-resolution, high-sensitivity SZ observations and their interpretation is the main challenge for future SZ parameter estimation and in the reconstruction of the cluster’s temperature and velocity structure.

Armed with these tools, we address a number of questions that are related to the effect of temperature and velocity variations on the SZ signal, with particular focus on parameter degeneracies and instrumental aspects. For example, we explicitly discuss the effect of angular resolution and frequency filters on the SZ signal, as well as different corrections to the location of the SZ null.
All these aspects, if ignored, lead to biases in the deduced cluster parameters.
We illustrate this for several examples, using both mock SZ measurements as well as recent SZ data.
We furthermore develop several tools for SZ parameter estimation which are now part of SZpack.
These should be useful for computing the SZ signal from cluster simulations and in the analysis of future high resolution, high sensitivity SZ measurements.

In this section we introduce the new temperature-velocity moment method to compute the SZ signal for general cluster atmospheres. This section is rather technical and mainly for readers interested in the computational details.

The SZ effect is caused by the scattering of CMB photons by moving electrons. For a small volume element of scattering electrons the SZ signal only depends on the electron temperature, Te, their total bulk velocity, βc, the direction cosine of this velocity with respect to the line-of-sight6, μc=^βc⋅^γ, and the Thomson optical depth of the scattering volume element, Δτ.
Both Te and Δτ are defined in the rest frame of the scattering volume element, while βc is defined with respect to the CMB rest frame.

For now, we shall assume that the observer is at rest in the CMB frame.
The conversion of the aforementioned parameters to the SZ signal can be expressed as ΔI(x)=ΔτFSZ(x,Te,βc,μc), where FSZ is a non-linear function and x=hν/kT0 with CMB temperature T0=2.726K(Fixsen et al., 1996; Fixsen & Mather, 2002).
The change in the CMB intensity can be further rewritten as

ΔI(x)

≈ΔτIox3[Y(x,Te)+β2cM(x,Te)

+βcP1(μc)D(x,Te)+β2cP2(μc)Q(x,Te)],

(1)

where Io=(2h/c2)(kT0/h)3≈270MJysr−1 and Pl(x) denote Legendre polynomials. The term Y(x,Te) describes the purely thermal SZ effect with temperature corrections included, while terms ∝βc are related to kinematic effects.

From previous analysis of the SZ effect, it is furthermore clear that the functions Y(x,Te), M(x,Te), D(x,Te) and Q(x,Te) can all be described using an appropriate set of frequency-dependent basis functions and temperature-dependent coefficients, where the latter encode the spatial dependence.
The integrated SZ signal along a given line-of-sight is therefore determined by appropriate moments of the cluster’s temperature and velocity field. These contain the desired information about the cluster gas and structure; as such they define the observables of the SZ measurement, and the aim will be to use the moments to learn about the cluster gas.

2.1 Low temperature gas (kTe≲10keV)

As discussed in Chluba et al. (2012), for electron gas temperatures kTe≲5keV−10keV an asymptotic expansion of the Boltzmann collision term (similar to Challinor & Lasenby, 1998; Itoh et al., 1998; Sazonov & Sunyaev, 1998; Nozawa et al., 1998a; Challinor & Lasenby, 1999) can be used to represent the SZ signal with high precision.
From Eq. (25) of CNSN, up to some specified correction order, kmax, in the electron temperature, it therefore follows (see Appendix A for more details):

Ylow(x,Te)

=kmax∑k=0Yk(x)θk+1e

(2a)

Mlow(x,Te)

=13M(x)+kmax∑k=0Mlowk(x)θk+1e

(2b)

Dlow(x,Te)

=G(x)+kmax∑k=0Dlowk(x)θk+1e

(2c)

Qlow(x,Te)

=1130Q(x)+kmax∑k=0Qlowk(x)θk+1e,

(2d)

with θe=kTe/mec2 , M(x)=Y0(x)+G(x), G(x)=xex/[ex−1]2, and Q=xGcoth(x/2).
The functions Yk are defined as in CNSN, while Mlowk, Dlowk, and Qlowk are given by Eq. (A).
To give an example, Y0=Q−4G describes the usual (non-relativistic) thSZ effect (Zeldovich & Sunyaev, 1969), while the term ∝G is related to the kSZ effect (Sunyaev & Zeldovich, 1980).
Both M(x) and Q(x) describe the lowest order kinematic terms ∝β2c.

For a fixed line-of-sight through the cluster medium, the total SZ signal is determined by integration over Δτ. With the decomposition given above it is convenient to introduce the following line-of-sight temperature and velocity moments:

y(k)

=∫θk+1edτ,

b(k)0

=∫β2cθkedτ,

(3)

b(k)1

=∫βcP1(μc)θkedτ,

b(k)2

=∫β2cP2(μc)θkedτ,

where the integrals are carried out in the cluster frame, with the condition Te≤Te,low for k>0 and for y(0).
Here y(k) denotes the generalized y-parameters, while b(k)i take into account the effect of the clusters global and internal gas motion. For example, y(0)=∫θedτ is the usual line-of-sight y-parameter or average thermal pressure of the electrons, while b(0)1 (no temperature dependence) is proportional to the average velocity of the cluster medium along the line-of-sight weighted by the electron number density.
The optimal values for Te,low and kmax depend on the required precision and will be specified below (see Sect. 2.4).
With Eq. (3) we can now define the moment vector

where we arranged the entries of \boldmathm\unboldmathlow with respect to orders in βc. Notice that the dimensions of the vectors \boldmathb\unboldmathi in principle can differ from kmax+1.
In particular, for the velocity moments usually fewer terms in the electron temperature are required to describe the SZ signal accurately, since they only lead to very small corrections (e.g., see Chluba et al., 2012).

If we assume that the SZ signal is observed at m frequencies, {xi}, we can furthermore introduce the signal vector for the contribution of the low temperature gas, \boldmathS\unboldmathTlow=(ΔIlow(x1),...,ΔIlow(xm)). This defines the matrix equation

with \boldmathY\unboldmathTlow=(Y0,...,Ykmax),
\boldmathD\unboldmathTlow=(Dlow0,...,Dlowkmax), and so on.
For the ith row of Flow the frequency-dependent functions evidently have to be evaluated at the required xi.
The SZ signal caused by low temperature gas can therefore be computed as a simple matrix operation once the low temperature-velocity moment vector, \boldmathm\unboldmathlow, is known. The columns of the moment matrix, Flow, are the values of the basis functions at the required frequencies.

2.2 High temperature gas (kTe≳10keV)

For gas with temperatures kTe≳10keV the convergence of the asymptotic expansion given above becomes slow (see Fig. 2). However, recently CNSN found an alternative set of frequency-dependent basis functions that allow very accurate description of the SZ signal up to high temperatures and bulk velocities.
The convergence radius of the CNSN expansion around the chosen pivot temperature (in CNSN kTe,0≃15keV was used) is also limited (see Fig. 2), but in combination with the low temperature expansion it allows covering a large part of parameter space.

Like for the asymptotic expansion, the signal is determined by particular temperature and velocity moments, but this time the weighting differs slightly from those of Eq. (3).
With the expressions given in CNSN it is straightforward to show (see Appendix B for more details) that up to some specified order, k∗max, of the electron temperature one has

Yhigh(x,Te)

=k∗max∑k=0Yhighk(x)N(θe)θke

(7a)

Mhigh(x,Te)

=k∗max∑k=0Mhighk(x)N(θe)θke

(7b)

Dhigh(x,Te)

=k∗max∑k=0Dhighk(x)N(θe)θke

(7c)

Qhigh(x,Te)

=k∗max∑k=0Qhighk(x)N(θe)θke,

(7d)

with N(θe)=e−1/θeK2(1/θe)θe≈4π(2πθe)3/2[1−152θe+345128θ2e+O(θ3e)].
Here, K2(x) denotes the modified Bessel functions of second kind.
The functions Yhighk, Mhighk, Dhighk, and Qhighk are defined by Eq. (B).
All temperature-independent kinematic terms were already taken into account by Eq. (2), so that they do not reappear here.
Also, in general k∗max≠kmax, although below we usually set k∗max≡kmax.

In analogy to the low temperature gas case we introduce the following line-of-sight temperature and velocity moments:

z(k)

=∫N(θe)θkedτ,

c(k)0

=∫β2cN(θe)θkedτ,

(8)

c(k)1

=∫βcP1(μc)N(θe)θkedτ,

c(k)2

=∫β2cP2(μc)N(θe)θkedτ,

where the integrals are carried out in the cluster frame, with the condition Te,low≤Te≤Te,high.
The optimal value for Te,high depends on the required precision. Also, one can split the temperature range up into different parts, each with their own set of basis functions defined on the intervals Te∈(Ti−1e,high,Tie,high], as will be
specified in Sect. 2.4.
With this we define the moment vector

with \boldmathY\unboldmathThigh=(Yhigh0,...,Yhighk∗max), \boldmathD\unboldmathThigh=(Dhigh0,...,Dlowk∗max), and so on, all as above.
Again the SZ signal caused by high temperature gas can be expressed as a simple matrix multiplication, once the moments are determined.
This reduces the computational burden to calculation of the temperature-velocity moment vector which solely depends on the cluster atmosphere.

2.3 Total line-of-sight SZ signal

With the definitions of the previous sections, the total SZ signal is given by \boldmathS\unboldmath=\boldmathS\unboldmathlow+\boldmathS\unboldmathhigh.
Introducing the total cluster temperature-velocity moment vector, \boldmathm\unboldmathT=(\boldmathm\unboldmathTlow,\boldmathm\unboldmathThigh), and the frequency-dependent matrix, Extra open brace or missing close brace, one has

\boldmathS\unboldmath=F%
\boldmathm\unboldmath.

(12)

It is clear that the dimension of S defines the maximal number of moments that could possibly be deduced from the SZ data.
However, in the presence of noise, foregrounds, and correlations between the moments, one naturally has dim(\boldmathm\unboldmath)<dim(\boldmathS%
\unboldmath).

It is furthermore important that for a given experimental precision, an optimal combination of the basis functions can be found which minimizes the number of moments required to accurately represent the SZ signal.
In particular, with the approach of CNSN one can vary the reference/pivot temperature, Te,0, and number of reference points used in the computation of the basis functions to improve the temperature coverage of the approximation.
In that case the moment vector can be cast into the form \boldmathm\unboldmathT=(\boldmathm\unboldmathTlow,\boldmathm\unboldmathThighI,\boldmathm%
\unboldmathThighII,...) with different temperature regions, [0,Te,low], (Te,low,TIe,high], (TIe,high,TIIe,high], (TIIe,high,TIIIe,high], and so on.
Here the subscript ‘low’ is used to indicate that the asymptotic expansion is applied for those moments, while for any moments with subscript ‘high’ we formulate the basis using CNSN.
We will discuss the associated optimization problem in Sect. 2.4.

With the formulation given above it is also straightforward to include the effect of angular resolution and frequency bands on the SZ signal.
The effect of angular resolution is accounted for by spatially averaging the temperature-velocity moments, i.e., \boldmathm\unboldmath→⟨\boldmathm%
\unboldmath⟩, where ⟨...⟩ denotes angular/spatial average.
The bandpass can be taken into account by means of a matrix W.
With this the SZ signal in more general can be expressed as

where we also added noise to the problem. In a similar way possible contaminations by (spatially) smooth foregrounds, radio sources, or dusty-star-forming galaxies (DSFGs) can be incorporated.

One of the benefits of the moment method described here is that for a given set of frequencies the moment matrix only has to be computed once.
This, for example, makes computation of the SZ signal from simulated cluster very efficient and accurate.
However, Eq. (12) is still mainly interesting from the computational point of view because the entries of the moment vector are not independent. For instance, all moments related to y(k) are non-negative and one also expects y(k)>y(k+1).
This imposes rather complicated priors and correlations among the different entries of the moment vector with the actual dimensionality of the problem being much smaller. We will show below that for future SZ observations only a few parameters are required to accurately describe the SZ signal, although the number of entries in the moment vector, m, is much larger.

Finally, we mention that the effect of the observers motion with respect to the cluster (Chluba et al., 2005, 2012) can be included using simple Lorentz-transformation of the frequencies and corresponding angles into the CMB rest frame, to account for the effect of Doppler boosting and relativistic light aberration (e.g., Chluba, 2011). However, for the discussion below these aspects of the problem are not crucial and will be omitted.

Figure 1: Deviation of the approximation from the numerical result. The asymptotic expansion is used to represent the SZ signal. For each of the curves, k+1 temperature orders were included. The departure is expressed in units of ΔIs≃0.013MJysr−1 and τ=0.01 was assumed.

Figure 2: Range of convergence for different approximations. The upper panel shows the results obtained with the asymptotic expansion, while for the lower panel the basis CNSN with θe,0=0.03 was used. The maximal departure for 0.1≤x≤30 is expressed in units of ΔIs≃0.013MJysr−1 and τ=0.01 was assumed for all cases.

2.4 Minimizing the required number of moments

One of the important questions is how many moments are needed to describe the SZ signal accurately for a given experimental sensitivity and range of gas temperatures.
Here it is particularly interesting to try minimizing the total number of moments that are required to achieve an optimal representation of the SZ signal.
To answer this question we first define a fiducial sensitivity for comparison. We shall use the kSZ signal of a cluster with line-of-sight optical depth τ=10−2 and βcμc=10−3 close to the thSZ crossover frequency νc=217GHz(xc=3.83) as benchmark; this gives a distortion with amplitude ΔIs≃9.76τβcμc(kT0)3/[h2c2]sr−1≃0.013MJysr−1.
For an isothermal cluster with kTe≃5keV electrons and τ=10−2 the maximal thSZ signal is roughly ΔIth≃0.17MJysr−1 at x≃6.7.
Therefore, ΔIs corresponds to ≃8% precision on ΔIth.

Before carrying out additional computations we extended the basis of CNSN with additional pivot points (at θe,0=0.01 and 0.1) such that the SZ signal can be represented in a wider range of temperatures.
We provide this basis both in the cluster rest frame and the CMB frame.
With the current version of SZpack a ≃0.001% precision is achieved at frequencies 0.01≲x≲30, temperatures kTe≲75keV, and for βc≲0.01 basically at no additional computational cost.
We furthermore included the necessary database directly into SZpack such that no time is consumed loading data.
In the current implementation evaluation of the SZ signal at 400 frequencies takes about 0.01 seconds on a standard laptop.
These routines can also be directly invoked from Python.

One can now calculate how accurately the different sets of basis functions describe the SZ signal for varying kmax.
For the asymptotic expansion we show two examples in Fig. 1. At low temperatures (upper panel) only a few terms in the expansion are needed to achieve a very accurate representation of the SZ signal. The mismatch is usually largest at high frequencies, x≃10, while below the crossover frequency higher order temperature terms are small, even for larger electron temperatures.
The lower panel of Fig. 1 indicates that at higher temperatures the convergence of the asymptotic expansion becomes slower, a problem that is well-known from previous analysis (e.g., see Itoh et al., 1998).

A simple calculation can be used to further quantify the convergence rate of the different basis functions:
for a given order in temperature we compute the maximal deviation of the approximation from the numerical result in the frequency range 0.1≤x≤30.
For both the asymptotic expansion and the basis CNSN with reference temperature θe,0=0.03 the results are shown in Fig. 2. One can see that for the asymptotic expansion the agreement with the numerical result does not improve above kTe≃13keV.
At higher temperatures the expansion of CNSN performs much better.
In particular, the convergence radius increases strongly when including the first few temperature corrections.
One can also observe that for the CNSN basis functions with pivot temperature θe,0=0.03 convergence above kTe≃40keV is not achieved. However, this can be overcome by adding another set of basis functions with pivot θe,0>0.03.
For fixed temperature correction order, kmax, one can therefore try to find an optimal combination of pivot temperatures to cover a large range of temperatures.
This is not the absolute minimum with respect to the number of moments, but optimization in this way still is beneficial while remaining sufficiently simple.

ΔI/ΔIs

kmax/ktot

Te,low

TIe,high/TIe,0

TIIe,high/TIIe,0

TIIIe,high/TIIIe,0

[keV]

[keV]

[keV]

[keV]

1

2/12

9.1

21.6/14

42/30

75/55

1

3/12

12.5

48.5/25

90∗/80

–

1

4/10

14.3

75/35

–

–

0.1

3/16

6.8

17.25/11

36.4/25

68/50

0.1

4/15

9.3

33.1/18.5

80∗/55

–

0.1

5/18

10.76

48.5/25

90∗/80

–

0.01

4/20

5.76

14.7/9.4

32/22

61/45

0.01

5/18

7.3

23.86/14

61/40

–

0.01

6/21

8.55

33.3/18.5

80∗/60

–

5×10−4

6/28

5.76

14.7/9.4

32/22

63.5/45

Table 1: Optimal distribution of temperature pivots, Tie,0, and regions, [0,Te,low], (Te,low,TIe,high], (TIe,high,TIIe,high], etc, for given accuracy goal, ΔI/ΔIs. At temperature Te≤Te,low the asymptotic expansion is used, while above expressions based on CNSN are applied.
As fiducial accuracy value we used ΔIs≃0.013MJysr−1 and optical depth τ=0.01. Numbers marked with asterisk are only estimated upper bounds, although the approximation is much better. The number of temperature correction terms in each region is kmax, while ktot gives the total number of temperature terms for all regions. The required number of temperature-velocity moments depends on the settings for kinematic corrections and is not further specified here.

In Table 1 we summarize the results of our efforts to cover at least the temperature range 0≤kTe≲60keV for a given precision and kmax.
We defined different regions of temperatures making sure that close to the boundaries the condition on the precision is met with some 10%−20% margin.
Far away from the boundaries of the different temperature regions the approximations are typically much more accurate.
The setting for accuracy goal 5×10−4ΔIs is already close to the numerical precision of our approximations and is mainly meant to provide an extreme setting for comparisons.
Furthermore, if the electron temperature is smaller than some maximal temperature, Te,max, the total number of required variables can be further reduced by dropping moments in regions with Te>Te,max.
For a given accuracy goal this defines an optimal value for kmax.
We also found that the same settings work when 0≲βc≲0.01.

One point we mention is that the settings given in Table 1 are in fact independent of the chosen optical depth, τ.
This means that only the scaling of the approximation with electron temperature affects the precision. For instance, if the optical depth along a given line-of-sight is τ≃2×10−3, but the temperature is fixed, then the absolute precision of the approximation at accuracy goal in the second category (denoted with 0.1ΔIs) is actually ≲0.02ΔIs.

To demonstrate how to use and interpret the temperature-velocity moment method we
now discuss the SZ signals for different cluster models. We start with the
simplest case of an isothermal cluster and then work our way through
several instructive examples, also introducing the simpler moment method that is applicable to sufficiently smooth (low temperature-velocity variance) cluster atmospheres.

3.1 SZ signals for isothermal clusters

where Ne,0≃10−3cm−3 is the typical central number density of free electrons, rc≃100kpc is the typical core radius of clusters, and β≃2/3(e.g., see Reese et al., 2002).
In the absence of bulk velocities, one therefore has the temperature moments y(k)=θk+1eτ for Te≤Te,low and z(k)=N(θe)θkeτ for Te>Te,low.
This shows that the spatial morphology of the SZ signal is fully determined by the overall optical depth factor, τ(^γ)(we shall ignore small corrections caused by multiple scattering, e.g., see Dolgov et al., 2001; Itoh et al., 2001; Colafrancesco et al., 2003), with the same spectral shape for any line-of-sight through the cluster.
This also implies that the average SZ signal measured for an unresolved cluster in this case is determined by only one spectral function, and the spatially averaged optical depth, ⟨τ⟩.

Allowing the cluster to move with a peculiar velocity, βc, relative to the CMB one readily obtains the velocity moments7b(k)0=β2cθkeτ, b(k)1≈βcμcθkeτ and b(k)2≈β2cP2(μc)θkeτ for clusters with Te≤Te,low, and c(k)0=β2cN(θe)θkeτ, c(k)1≈βcμcN(θe)θkeτ, and c(k)2≈β2cP2(μc)N(θe)θkeτ for hot clusters.
Again the spatial dependence of the SZ signal factors out and is determined by the one of the line-of-sight optical depth alone.
It is however clear that the SZ morphology becomes frequency-dependent8 once the cluster no longer is isothermal or significant internal motions of the ICM are present.
The reason for this frequency-dependence is related to the variation of the temperature-velocity moments along different lines-of-sight, as we illustrate below (Sect. 3.4).

3.2 SZ signals for smooth density and temperature profiles

In more realistic cluster models, the variation of the temperature also has to be included.
One common possibility assumes a polytropic temperature profile, Te∝ρ1−γgas(Markevitch et al., 1999; Finoguenov et al., 2001; Pratt & Arnaud, 2002).
Alternatively, one can consider fits to the observed temperature and density profiles derived from Chandra X-ray data (Vikhlinin et al., 2006).

For our discussion it is only important that the associated profiles are very smooth. This suggests that a good approximation for the SZ signal can be found by computing average values for the temperature and velocity along the line-of-sight.
Corrections to this lowest order approximation can then be included using a Taylor-series around the average values.
Assuming that βc=0, we can introduce the SZ-weighted electron temperature,

Te,SZ(^γ)

=(mec2/k)y(0)/y(−1)=τ−1∫Tedτ.

(15)

The integrals y(k) are defined by Eq. (3) but here we do not impose any condition on the electron temperature.
One can furthermore introduce the isothermal temperature moments

y(k)iso(^γ)=[kTe,SZ(^γ)/mec2]k+1τ(^γ).

(16)

In general these moments are not identical to y(k)(^γ) for k>0, and the ratio ρ(k)(^γ)=y(k)/y(k)iso≡τky(k)/[y(0)]k+1
can be used to quantify departures from isothermality:
in regions with ρ(k)≠1 one expects the SZ signal to be poorly represented by just using the SZ-weighted electron temperature and line-of-sight optical depth.
The ratios ρ(k) are also independent of the overall temperature and density scales. They only depend on the shapes of the cluster temperature and electron density profiles.

We can write this more formally by using the isothermal moment vector, \boldmathm\unboldmathiso(τ,Te,SZ), and expanding the average SZ signal around Te,SZ and τ:

where the first order derivative term canceled after performing the line-of-sight average.
Here \boldmathS\unboldmathiso(τ,Te,SZ)=F\boldmathm\unboldmathiso(τ,Te,SZ) is the leading order, average SZ signal, while ∂kθe\boldmathS\unboldmathiso(Te)=F∂kθe\boldmathm%
\unboldmathiso(Te) is the derivatives of the SZ signal with respect to θe.
To simplify the notation, we furthermore introduce

Δy(k)

=∫(θe−θe,SZ)k+1dτ=y(k)isok+1∑m=0(k+1m)(−1)k+1−mρ(m−1).

This means Δy(0)=0, Δy(1)=y(1)−y(1)iso=y(1)isoΔρ(1), Δy(2)=y(2)isoΔρ(2)−3y(2)isoΔρ(1), and so on, with Δρ(k)=ρ(k)−1.
Defining ω(k)=Δy(k)/y(k)iso≡τkΔy(k)/[y(0)]k+1 we then can finally write

where \boldmathS\unboldmath(k)iso(τ,T)=(Tk/k!)∂kT\boldmathS\unboldmathiso(τ,T).
In this parametrization the observables for the SZ measurement are τ, Te,SZ, and the temperature moments ω(k).

With SZpack it is straightforward to compute the required vectors, \boldmathS\unboldmath(k)iso, with very high precision.
In Fig. 3 we show the first few \boldmathS\unboldmath(k)iso.

Figure 3: Spectral functions \boldmathS\unboldmath(k)iso in units of τ and S0=(2h/c2)(kT0/h)3≈270MJysr−1. For comparison we also show the main SZ signal (solid black line). The arrow indicates the direction of increasing temperature, with the lines within the groups being separated by ΔTe=10keV.

The typical amplitude of the \boldmathS\unboldmath(k)iso is dropping with k, indicating that unless the temperature moments ω(k) increase strongly with k, higher order terms remain small with the largest contributions at high frequencies.
This shows that the considered expansion becomes fully perturbative unless rather large deviations from the smooth temperature profile case are present.

For example, using the simple fits for one of the hottest clusters (Te≃9.2keV at r≃330kpc), A2029, from the cluster sample of Vikhlinin et al. (2006), we find ω(1)≃0.16, ω(2)≃0, and ω(3)≃0.05 close to the cluster center.
This indicates that higher order corrections decay rapidly.
In fact, the correction related to ω(1) contributes at the level of a few percent to the average SZ signal, while higher order moments are negligible.
We find that even for more realistic cases from simulated clusters only a few moments of the temperature field need to be known to accurately describe the SZ signal (Sect. 3.4).
Also, in the more extreme case of a two-temperature plasma only the first few terms are required (Sect. 3.3).

Some interesting frequencies for the functions \boldmathS\unboldmath(k)iso are related to their nulls, maxima and minima. For \boldmathS\unboldmath(1)iso (which is not shown in Fig. 3) we find a rather temperature-independent minimum at x≃2.26. At x≃4 it crosses zero for Te≃10keV, while for Te≃50keV the null is located at x≃4.89. Its maximum is located at x≃7.2 for Te=10keV and at x≃10 for Te=50keV.
For \boldmathS\unboldmath(2)iso (see Fig. 3) on the other hand we find a maximum at x≃2.12 and the first null at x≃3.5.
The position of the minimum varies from x≃5.8 for Te=10keV to x≃7.2 for Te=50keV.
These properties might be useful when deciding about the locations of frequency channels in future SZ experiments.

We emphasize that for the computation of \boldmathS\unboldmath(0)iso(τ,Te,SZ) and \boldmathS\unboldmath(2)iso(τ,Te,SZ) a large number of temperature terms has to be included.
Although in the example given above we only find ω(1) to be significant as additional parameter, this is not equivalent to dropping higher order temperature terms.
This point is very important when interpreting future SZ data, since otherwise biased results for τ,Te,SZ, and ω(1) are obtained.
Similarly, one has to include βcμc≠0 for the analysis, as we discuss in more detail below.

We also note that Eq. (17) is applicable even if the real temperature distribution is not a smooth function. It is only important that the variance of the temperature and thermal pressure remains sufficiently low to warrant decreasing values of the moments, ω(k), with larger k. This condition is usually fulfilled even in more realistic cluster models (see Sect. 3.4).

The effect of line-of-sight temperature variance

Above we showed that the dominant correction to the SZ signal is determined by the temperature moment ω(1). This parameter can be interpreted as line-of-sight variance or dispersion of the electron temperature but weighted by the optical depth of the scattering volume element.
One can now address the question of how important this term is for the interpretation of the SZ signal. In particular, by how much are the deduced best-fit values for Te,SZ and τ affected if the contribution from ω(1) is neglected for high-sensitivity, multi-frequency SZ measurements.

We can start by writing the SZ signal for T∗e,SZ≠Te,SZ as an expansion of \boldmathS\unboldmath(τ∗,T∗e,SZ) around Te,SZ and τ:

where Θ=(T∗e,SZ−Te,SZ)/Te,SZ, Θτ=(τ∗−τ)/τ, and we neglected higher order terms.
To determine the best-fit values for τ∗ and T∗e,SZ one has to compare to \boldmathS\unboldmath(τ,Te,SZ,ω(1))≈%
\boldmathS\unboldmath(0)iso(τ,Te,SZ)+\boldmathS\unboldmath(2)iso(τ,Te,SZ)ω(1) and then minimize the squared difference.
The coefficients relating Θτ and Θ to ω(1) then are only functions of temperature, and it is straightforward to compute the degeneracy coefficients ατ=−Θτ/ω(1) and αT=Θ/ω(1) (Fig. 4).
Both τ and Te,SZ are correlated with ω(1) to a similar degree.
The degeneracy is very close to unity at low temperatures and only drops to about 1/2 at very high temperatures.
We find that ατ(Te)≈[1+2.7×10−2T0.86e]−1 and αT≈exp(−2.6×10−2T0.86e) match the full numerical result with ≃10% precision.
With these expressions we can directly estimate the expected value for T∗e,SZ obtained by computing the best-fits to the full SZ signal.

Figure 4: Degeneracies of τ and Te,SZ with ω(1). The expected (biased) best-fit values are τ∗≈τ[1−ατ(Te)ω(1)] and T∗e,SZ≈Te,SZ[1+αT(Te)ω(1)] when analyzing the SZ signal.

To determine the degeneracy coefficients we used a very dense grid of frequency points in the range x=0.1 to 30. More realistically far fewer independent frequency bins are available plus the signal is averaged over some bandwidth and the beam. Furthermore, foregrounds and the experimental sensitivity at each frequency are important.
All these aspects affect the degeneracy between the SZ parameters, as we explain in more detail below (Sect. 5.1).
Nevertheless, the estimate obtained above gives a rough scaling for the importance of line-of-sight temperature variations for the interpretation of the SZ measurement.

The effect of velocity terms on the SZ signal

Thus far we neglected the effect of bulk velocity on the SZ signal. However, the effect of (internal) motions can again be included by expanding the SZ signal around the mean line-of-sight values.
We first define the two velocity components βc,∥=βcμc and βc,⊥=βc√1−μ